Average Error: 18.5 → 1.4
Time: 20.4s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{-v}{t1 + u}}{\frac{t1 + u}{t1}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{-v}{t1 + u}}{\frac{t1 + u}{t1}}
double f(double u, double v, double t1) {
        double r26831 = t1;
        double r26832 = -r26831;
        double r26833 = v;
        double r26834 = r26832 * r26833;
        double r26835 = u;
        double r26836 = r26831 + r26835;
        double r26837 = r26836 * r26836;
        double r26838 = r26834 / r26837;
        return r26838;
}
↓
double f(double u, double v, double t1) {
        double r26839 = v;
        double r26840 = -r26839;
        double r26841 = t1;
        double r26842 = u;
        double r26843 = r26841 + r26842;
        double r26844 = r26840 / r26843;
        double r26845 = r26843 / r26841;
        double r26846 = r26844 / r26845;
        return r26846;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.5

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  2. Using strategy rm

  3. Applied times-frac1.3

    \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
  4. Using strategy rm

  5. Applied neg-mul-11.3

    \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \cdot \frac{v}{t1 + u}\]

  6. Applied associate-/l*1.4

    \[\leadsto \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \cdot \frac{v}{t1 + u}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity1.4

    \[\leadsto \frac{-1}{\frac{t1 + u}{\color{blue}{1 \cdot t1}}} \cdot \frac{v}{t1 + u}\]

  9. Applied *-un-lft-identity1.4

    \[\leadsto \frac{-1}{\frac{\color{blue}{1 \cdot \left(t1 + u\right)}}{1 \cdot t1}} \cdot \frac{v}{t1 + u}\]

  10. Applied times-frac1.4

    \[\leadsto \frac{-1}{\color{blue}{\frac{1}{1} \cdot \frac{t1 + u}{t1}}} \cdot \frac{v}{t1 + u}\]

  11. Applied *-un-lft-identity1.4

    \[\leadsto \frac{\color{blue}{1 \cdot -1}}{\frac{1}{1} \cdot \frac{t1 + u}{t1}} \cdot \frac{v}{t1 + u}\]

  12. Applied times-frac1.4

    \[\leadsto \color{blue}{\left(\frac{1}{\frac{1}{1}} \cdot \frac{-1}{\frac{t1 + u}{t1}}\right)} \cdot \frac{v}{t1 + u}\]

  13. Applied associate-*l*1.4

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{1}} \cdot \left(\frac{-1}{\frac{t1 + u}{t1}} \cdot \frac{v}{t1 + u}\right)}\]

  14. Simplified1.4

    \[\leadsto \frac{1}{\frac{1}{1}} \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{\frac{t1 + u}{t1}}}\]

  15. Final simplification1.4

    \[\leadsto \frac{\frac{-v}{t1 + u}}{\frac{t1 + u}{t1}}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))